{"id":"n-7036cf2590641970","revision":0,"first_index":0,"fingerprint":"f6-700e7903acaa52c2","name":"one_plus_derivative of veronese_section_2_1 of Even-numbered rows of Pascal's triangle.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":-1,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[[],["2"],["8","4"],["26","32","6"],["64","168","80","8"],["130","624","612","160","10"],["232","1804","3168","1672","280","12"],["378","4368","12298","11440","3822","448","14"],["576","9296","38688","57200","33280","7728","672","16"],["834","17952","104040","226304","207740","83232","14280","960","18"],["1160","32148","248064","749360","1007760","633080","186048","24624","1320","20"],["1562","54208","537966","2160224","4024580","3689952","1691228","381216","40194","1760","22"],["2048","87032","1080816","5576120","13788224","17473008","11648672","4076336","728640","62744","2288","24"],["2626","134160","2039180","13156000","41755350","70129760","64483720","32722560","9044750","1315600","94380","2912","26"],["3304","199836","3650400","28811640","114277800","246127140","299537280","209235600","83636280","18747300","2265120","137592","3640","28"]],"degrees":[-1,0,1,2,3,4,5,6,7,8,9,10,11,12,13],"row_sums":["0","2","12","64","320","1536","7168","32768","147456","655360","2883584","12582912","54525952","234881024","1006632960"],"nonn":true,"definitions":[{"position":0,"source":"derived:one_plus_derivative","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["2"]],"recurrence":{"denominator":{"coeffs":[["-9","11","-2"],["16","-15","2"],["-9","4","0"],["2","0","0"]]},"terms":[{"coeff":{"coeffs":[["0","-16","36","-8"],["4","23","-41","6"],["-3","-21","12","0"],["2","6","0","0"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","6","-20","18","-4"],["0","-8","16","-8","0"],["0","4","-4","0","0"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":175509,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-41a8aca04e899131"]},{"id":175510,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-41a8aca04e899131"]},{"id":175511,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-41a8aca04e899131"]}],"region_exact":{},"incoming_edges":[{"id":"e-41a8aca04e899131","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"derivative_polynomial\",\"coefficients\":[{\"numerator\":[\"1\"],\"denominator\":[\"1\"]},{\"numerator\":[\"1\"],\"denominator\":[\"1\"]}]}"},"operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-7036cf2590641970","premise_count":1,"theorem":"Laguerre's theorem on differential operators: if phi is a real polynomial with only real roots, then phi(D) maps every real-rooted polynomial to a real-rooted polynomial or zero (write phi(D) = C D^j prod (1 + c_i D); D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"c3fc2af3bfc538189438acae0323c31226890318871bc4717d88de54e87e3384","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-7a03fdbbc6f038d4","role":"sequence","name":"veronese_section_2_1 of Even-numbered rows of Pascal's triangle.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-4cbb9033375e56c6","kind":"fingerprint_related","node_a_id":"n-f7c96885eeb8238b","node_b_id":"n-7036cf2590641970","fingerprint":"f6-700e7903acaa52c2","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-700e7903acaa52c2"},"other_id":"n-f7c96885eeb8238b","other":{"id":"n-f7c96885eeb8238b","name":"one_plus_derivative of One half of odd-numbered entries of even-numbered rows of Pascal's triangle A007318.","first_index":0,"fingerprint":"f6-700e7903acaa52c2","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":0,"preview_rows":[["1"],["4","2"],["13","16","3"]],"card_rows":[["1"],["4","2"],["13","16","3"],["32","84","40","4"]],"row_sums_preview":["1","6","32","160","768","3584","16384","73728"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-2245b35ec7cd4857","steps":["D+1"],"edges":["e-b0514b9b0a774d8e"],"text":"A091044 \u2192 D+1"},"headline":"A091044 \u2192 D+1","chain":null}}],"oeis":[],"bfiles":[],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0:   0;\n  1:   2;\n  2:   8,     4;\n  3:  26,    32,      6;\n  4:  64,   168,     80,      8;\n  5: 130,   624,    612,    160,     10;\n  6: 232,  1804,   3168,   1672,    280,    12;\n  7: 378,  4368,  12298,  11440,   3822,   448,    14;\n  8: 576,  9296,  38688,  57200,  33280,  7728,   672,  16;\n  9: 834, 17952, 104040, 226304, 207740, 83232, 14280, 960, 18;\n","recurrence_sentence":"The row polynomials satisfy (2*n^3-9*n^2+16*n-9+(4*n^2-15*n+11)*x+(2*n-2)*x^2)*P(n,x) = (2*n^3-3*n^2+4*n+(6*n^3-21*n^2+23*n-16)*x+(12*n^2-41*n+36)*x^2+(6*n-8)*x^3)*P(n-1,x) + ((4*n^2-8*n+6)*x-(4*n^2-16*n+20)*x^2-(8*n-18)*x^3-4*x^4)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 2.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 2;\nP[n_, x_] := P[n, x] = Expand[Together[((2*n^3 - 3*n^2 + 4*n + (6*n^3 - 21*n^2 + 23*n - 16)*x + (12*n^2 - 41*n + 36)*x^2 + (6*n - 8)*x^3)*P[n - 1, x] + ((4*n^2 - 8*n + 6)*x - (4*n^2 - 16*n + 20)*x^2 - (8*n - 18)*x^3 - 4*x^4)*D[P[n - 1, x], x])/(2*n^3 - 9*n^2 + 16*n - 9 + (4*n^2 - 15*n + 11)*x + (2*n - 2)*x^2)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 2; for(n = 2, N, my(i = n + 1); v[i] = ((2*n^3 - 3*n^2 + 4*n + (6*n^3 - 21*n^2 + 23*n - 16)*x + (12*n^2 - 41*n + 36)*x^2 + (6*n - 8)*x^3)*v[i - 1] + ((4*n^2 - 8*n + 6)*x - (4*n^2 - 16*n + 20)*x^2 - (8*n - 18)*x^3 - 4*x^4)*deriv(v[i - 1], x))/(2*n^3 - 9*n^2 + 16*n - 9 + (4*n^2 - 15*n + 11)*x + (2*n - 2)*x^2)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-08f49539099949d5","steps":["Veronese 2 (j=1)","D+1"],"edges":["e-7b8f3a8fe3cbfab2","e-41a8aca04e899131"],"text":"A034870 \u2192 Veronese 2 (j=1) \u2192 D+1"},"headline":"A034870 \u2192 Veronese 2 (j=1) \u2192 D+1","chain":null}